Analyis and design of reset control systems = Análisis y diseño de sistemas de control reseteados

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UNIVERSIDAD DE MURCIA

FACULTAD DE INFORMÁTICA

Analysis and Design of Reset

Control Systems

Análisis y Diseño de Sistemas

de Control Reseteados

D. Miguel Ángel Davó Navarro

2015

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UNIVERSIDAD DE MURCIA

FACULTAD DE INFORMÁTICA

Análisis y Diseño de Sistemas de

Control Reseteados

Tesis Doctoral

Presentada por:

Miguel Ángel Davó Navarro

Supervisada por:

Dr. Alfonso Baños Torrico

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UNIVERSIDAD DE MURCIA

FACULTAD DE INFORMÁTICA

Analysis and Design of Reset

Control Systems

Ph.D. Thesis

Author:

Miguel Ángel Davó Navarro

Thesis Advisor:

Dr. Alfonso Baños Torrico

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Agradecimientos

En primer lugar me gustaría dar las gracias a mi director de tesis Alfonso Baños por guiarme durante estos cuatro años, compartiendo conmigo sus conocimientos. Agradecerle también el haber inculcado en mi un sentido de seriedad, responsabilidad y rigor académico, haciendo que me sienta orgulloso de mi trabajo. También agradecer a Aurelio Arenas por su disponibilidad y ayuda en los trabajos experimentales. Por supuesto agradecer a Joaquin Cervera su apoyo, reflexiones y consejos. También debo agradecer a los miembros de mi grupo de investigación, Angeles, Paco, Felix y al recién llegado Cristian por toda la ayuda que me han prestado haciendo mas fácil el camino hasta aquí. En especial agradecer a Pedro por ser un gran compañero y un buen amigo durante estos años, aguantando mis quejas y frustraciones, y ayudando siempre que fue necesario.

Agradecer a Jose Carlos por su colaboración durante la tesis, recibiéndonos con los brazos abiertos en Almería. Dar las gracias también a Antonio Barreiro y a los miembros de su grupo de investigación, Enma, Cesareo, Jose Antonio, Miguel y Pablo por su compañía en las reuniones de grupo, creando un ambiente de trabajo y colaboración perfectos.

También quiero mostrar mi gratitud a Sophie Tarbouriech, Frederic Gouaisbaut, Alexandre Seuret, William Heath y Joaquin Carrasco por acogerme en sus respectivos grupos de investigación, dándome la oportunidad de trabajar junto a ellos, enseñándome nuevos métodos que han ayudado a mi formación como investigador.

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sobrinos Elia, Pascual y Nayara, porque son mi mayor fuente de alegría. También dar las gracias a mi tío Jose Antonio por quererme como a un hijo y preocuparse siempre por mi bienestar. Importante también agradecer el apoyo de todos mis amigos, que siempre han estado a mi lado cuando les he necesitado.

Financiación

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Abstract

This thesis is focused on the field of reset control, which emerged more than 50 years ago with the main goal of overcoming the fundamental limitations of the linear com-pensators. In the last years, a multitude of works have shown the potential benefits of the reset compensation. Nevertheless, there are only a few works dealing with two of the most common limitations in industrial process control: the time-delay and the saturation. It is well-known that both limitations may lead into a detriment of the performance, and even the destabilization of the closed-loop system. Therefore, this thesis aim at analyzing the stability of the reset control systems under these limitations.

First, we address the stability analysis of time-delay reset control systems. We develop stability criteria for impulsive delay dynamical systems, based on the Lyapunov-Krasovskii method. The applicability of the results to time-delay reset control systems arises naturally, since they are a particular class of impulsive systems. We first focus our attention on impulsive delay systems with state-dependent resetting law. The application of the criteria to reset control systems supposes an improvement of the results in the literature, guaranteeing the stability of the system for larger values of the time-delay. In addition, in order to overcome the limitations of the base system, we consider the stabilization of impulsive delay systems by imposing time-dependent conditions on the reset intervals. As a result, stability criterion is developed for im-pulsive systems, and in particular time-delay reset control systems, with unstable base system. Finally, the new criterion is used to establish conditions for the stability of a reset control system containing a proportional-integral plus Clegg (PI+CI) integrator compensator.

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system by polytopes and directed graphs.

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Resumen

Esta tesis se enmarca principalmente en el area de los sistemas de control reseteados, con especial énfasis en el análisis de estabilidad y diseño de los sistemas de control compuestos de controladores reseteados. El control mediante controladores reseteados surgió hace 60 años con el claro objetivo de superar las limitaciones fundamentales de los controladores lineales. H. W. Bode demostró en 1940 que existe una relación entre la fase y la magnitud de la función de respuesta en frecuencia de un sistema lineal e invariante en el tiempo. Esta y otras propiedades han sido utilizadas desde entonces para caracterizar que es posible conseguir mediante un control lineal, es decir cuales son las limitaciones fundamentales de los controladores lineales. Un forma de superar estas limitaciones es buscar controladores no lineales que no estén sometidos a ellas. Esta idea es explotada por primera vez en 1958, cuando J. C. Clegg introduce un nuevo controlador no lineal, el cual consiste en un integrador cuyo estado es puesto a cero siempre que su entrada es igual a cero. Este integrador no lineal es la prime-ra propuesta de controlador reseteado y es conocido como Integprime-rador de Clegg. Un análisis del Integrador de Clegg mediante la función descriptiva revela un aumento de fase de 52◦ _{con respecto al integrador lineal, a consta de un pequeño aumento de la}

magnitud. Esta característica del Integrador de Clegg lo convierte en un potencial ele-mento para la superación de las limitaciones fundamentales de los controladores lineales.

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planta cuando los limites de la saturación son alcanzados. Este hecho puede suponer la aparición de ciclos limites, inestabilidad, etc. Por lo tanto, es imprescindible el desarrollo de herramientas de análisis de los sistemas de control reseteados en presencia de retardo y saturación. Por otra parte, hay dos factores importantes para que los controladores reseteados puedan ser satisfactoriamente aplicados en procesos industria-les. En primer lugar su diseño debe ser sencillo, de manera que puedan ser fácilmente comprendidos por los ingenieros de control e implementados en los sistemas de auto-matización industriales. En segundo lugar, es necesario desarrollar métodos sencillos que puedan ser utilizados para el ajuste de los parámetros de diseño de los controladores.

En este contexto, los objetivos principales de esta tesis son: el desarrollo de crite-rios que permitan garantizar la estabilidad de los sistemas de control recetados con retardo, caracterizar la región de convergencia asintótica de los sistemas de control reseteados en presencia de saturación a la entrada de la planta, desarrollar un método sistemático para el diseño de controladores reseteados, que permita cuando sea posible la obtención de reglas de sintonización sencillas, y finalmente mostrar los beneficios de los controladores reseteados y las reglas de ajuste propuestas mediante experimentos de control en varios procesos industriales.

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y aplicados a otros tipos de sistemas impulsivos.

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de estabilidad propuesto en esta tesis carece de dicha limitación, lo que permite su aplicación a este tipo de sistema de control reseteados.

Otro objetivo de la tesis es la caracterización de la región de atracción de los sistemas de control reseteados en presencia de saturación a la entrada de la planta. En los resultados previos se propone una aproximación elipsoidal de la región de atracción, la cual es obtenida mediante el segundo método de Lyapunov. Sin embargo, en esta tesis se propone una aproximación mediante la unión de politopos, que siendo no convexa proporciona mayor flexibilidad. La idea principal del método propuesto es la división del espacio de estados mediante politopos, y la utilización de un grafo dirigido cuyos nodos son los politopos y los enlaces entre los nodos representan las trayectorias del sistema fluyendo a través de los politopos. La representación mediante un grafo directo permite caracterizar de forma compacta el comportamiento del sistema de control reseteado. Además, se desarrollan un conjunto de resultados teóricos que garantizan la existencia y unicidad de las soluciones, y permiten comprobar la existencia de conjuntos invariantes en el interior de los politopos. Mediante estos resultados teóricos es posible determinar si las trayectorias del sistema convergen al origen para un determinado conjunto de condiciones iniciales, es decir una aproximación de la región de atracción. Con el objetivo de obtener estimaciones mas grandes se propone un algoritmo iterativo de mo-dificación del grafo mediante particiones de los politopos. Finalmente, varios ejemplos muestran que la estimación de la región de atracción obtenida mediante el procedi-miento propuesto es significativamente mayor que la obtenida con los resultados previos.

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mediante la respuesta impulsional de un conjunto de sistemas lineales e invariantes en el tiempo. El análisis y optimización del conjunto de respuestas impulsiones permite obtener un conjunto de reglas de sintonización sencillas aplicables a procesos modelados como sistemas de primer y segundo orden. Por otra parte, se contempla el caso de procesos modelados como integradores con retardo. En este caso, se desarrolla una regla de ajuste para el sistema base que permite garantizar una respuesta oscilato-ria, necesaria para que se produzcan acciones de reset. Además, se obtiene una regla de ajuste para el porcentaje de reset que permite mejorar del rechazo de perturbaciones.

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List of Figures

1.1 Standard feedback control system formed by a plant P and a compensator C. . . 2 1.2 Response of the Clegg integrator and the linear integrator to a sinusoidal

input. Input (dotted), linear integrator (dashed), and Clegg integrator (solid). . . 4 1.3 Historical view of the number of published works by year. . . 12 1.4 Distribution of the number of published works by countries. . . 12

2.1 Illustration of the distributed state. The shaded area is the domain of

xtk+τ. . . 37 2.2 Illustration of the shifted-distributed state representation. The shaded

areas are the domains of χk(0,·),χk(∆₂k,·), and χk(∆k,·). . . 37

2.3 Standard reset control system formed by a LTI plant P, a reset com-pensator R and two exosystems Σ1 and Σ2. . . 41

2.4 Standard time-delay reset control system formed by a LTI plant P, a reset compensator R, exosystems Σ1 and Σ2, and input/output time-delay. 47

2.5 Structure of the PI+CI compensator. . . 49 2.6 PI+CI describing function forkp = 1,τi = 1 and several values of the

reset ratio: pr= 0 (solid), pr = 0.5 (dashed), and pr = 1 (dotted). . . . 52

2.7 PI+CI describing function of the Clegg integrator with variable band resetting law and a first order low pass filter. Frequency is normalized by the parameterθ of the variable band. . . 55

2.8 Switching signals: time-triggered with ρ≈ts and triggered-by-counter

with Ns = 3. . . 58

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vi List of Figures

3.1 Trajectory and value of the LK functional for the IDDS with nonlinear and time-varying base system of the Example 3.1. . . 72 3.2 Trajectory of the reset control system and its base system from Example

3.4, with initial condition ϕ(θ) = (1,0), θ ∈[−1,0]. xp (dotted) andxr

(solid). . . 83 3.3 Trajectory and value of the LK functional (matrices obtained without

condition (3.58b)) for the time-delay reset control system of the Example 3.7 with initial condition ϕ(θ) = (1,0), θ ∈[−0.8,0]. . . 94

3.4 Notation example: r(1) = 1,r(2) = 2,r(3) = 2,N1 = 2,N2 = 1,N3 = 0, R(1) = 2, R(2) = 3, R(3) = 3. . . 98

3.5 Illustration of the evolution of a LK functional satisfying the conditions of Proposition 3.6. V (solid) and W (dotted). . . 100

3.6 Input time-delay: Trajectory and value of the LK functional V and

functional W (it is only defined after the reset instants) for the

time-delay reset control system of the Example 3.8 with input time-time-delay

h= 0.844, r = 2 and initial condition ϕ(θ) = (1,0), θ ∈[−0.844,0]. . . 108

3.7 Output time-delay: Trajectory and value of the LK functional V and

functional W (it is only defined after the reset instants) for the

time-delay reset control system of the Example 3.8 with output time-time-delay

h= 0.788, r = 2 and initial condition ϕ(θ) = (1,0), θ ∈[−0.788,0]. . . 108

3.8 Allowable reset interval as a function of the time-delay for the time-delay reset control system of the Example 3.10 obtained by Proposition 3.9 with r= 10. Input time-delay (solid) and output time-delay (dashed). . 114

3.9 Input time-delay: Trajectory and value of the LK functional and the functional W for the time-delay reset control system of the Example

3.10 with input time-delay h= 1.2, r= 10, ∆m = 0.121, ∆M = 0.5, and

initial conditionϕ(θ) = (1,0), θ∈[−1.2,0]. . . 115

3.10 Output time-delay: Trajectory and value of the LK functional and the functional W for the time-delay reset control system of the Example

3.10 with output time-delay h = 1.2, r = 10, ∆m = 0.121, ∆M = 0.5,

and initial condition ϕ(θ) = (1,0), θ∈[−1.2,0]. . . 116

3.11 Input delay: Zoom of the value of the LK functional for the time-delay reset control system of the Example 3.10. . . 117 3.12 Output time-delay: Zoom of the value of the LK functional for the

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List of Figures vii

3.13 Trajectory and value of the LK functional and the functionalW for the

IDDS of the Example 3.11 with time-delayh= 0.1,r= 40, ∆m = 0.0026,

∆M = 0.0671, and initial conditionϕ(θ) = (1,−1), θ ∈[−0.1,0]. . . 119

3.14 Allowable reset interval (periodic reset) as a function of the time-delay for the IDDS of the Example 3.12 obtained by Proposition 3.9 withr = 10.120

3.15 Trajectory and value of the LK functional and the functionalW for the

IDDS of the Example 3.12 with time-delay h= 1, r= 10, ∆m = ∆M =

0.6, and initial conditionϕ(θ) = (1,−1), θ ∈[−1,0]. . . 121

3.16 Allowable reset interval as a function of the time-delay for the time-delay reset control system of the Example 3.13 with β = 0.5, obtained by

Proposition 3.10 withr = 10. . . 127

3.17 Allowable reset period as a function of the time-delay for the time-delay reset control system of the Example 3.13 obtained by Proposition 3.10 with r = 10. . . 128

3.18 Trajectory and value of the LK functional and the functional W for

the time-delay reset control system of the Example 3.13 with β = 0.5,

time-delay h = 0.2, r = 10, ∆m = 0.001, ∆M = 0.978, and initial

condition ϕ(θ) = (1,0), θ ∈[−0.2,0]. . . 129

3.19 Trajectory of the time-delay reset control system of the Example 3.13 and its base system with time-delay h = 0.1, ∆m = 0.121, ∆M = 0.5,

and the initial condition ϕ(θ) = (1,0), θ ∈[−0.1,0]. . . 130

3.20 Trajectory of the time-delay reset control system of the Example 3.13 with time-delay h= 0.1 and initial conditionϕ(θ) = (1,0), θ ∈[−0.1,0].

Trajectory with reset period 0.1 (Fig. a), and trajectory with no evolution of the compensator states (Fig. b). . . 131

4.1 Reset control system with plant input saturation. . . 135 4.2 Illustration of the regions of saturation. Rj,j ∈ {1,2,3}. . . 136

4.3 Illustrative example of a state-space partition and its associated graph. 144 4.4 Connected polytopes: PI

∂i

−→ PF. . . 145

4.5 Illustrative example of the construction of the graph Dreset. All the

trajectories fromP1 trough the reset surface reach the subset R0C∩ MR

(red line). . . 146 4.6 Illustration of the four cases studied in the proof of Proposition 4.7. . . 150 4.7 Regions of saturation, Rj, j ∈ {1,2,3}. . . 155

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viii List of Figures

4.9 Obtained state-space partition and directed graph associated with the saturated reset control system. . . 157 4.10 Regions of saturation, Rj, j ∈ {1,2,3}. . . 159

4.11 Obtained state-space partition: Region of convergence: proposed method (blue and red) and Corollary 4.1 (red). . . 160

5.1 Frequency responses of the reset control system composed by a first order plant (a0 = 0.5, b0 = 1.5) and a PI+CI compensator (kp = 2,

τi = 0.15). . . 168

5.2 Output signals, control signals, and reset ratio of the reset control system and its base PI from the Example 5.1 for a sequence of step setpoints. PIbase (dotted-blue) and PI+CI (solid-black). . . 170

5.3 Output signals and control signals of the reset control system and its base PI from the Example 5.1 for a sequence of step disturbances. PIbase

(dotted-blue) and PI+CI (solid-black). . . 171 5.4 Output signals, control signals and reset ratios of the reset control

system from the Example 5.1 with a 20% of uncertainty in the plant gain. k= 2.4 (dashed),k = 3 (solid), and k = 3.6 (dotted). . . 172

5.5 Illustration of the error signals sequence with zero crossing resetting law.174 5.6 Frequency responses of the reset control system composed by a second

order plant (a1 = 6,a0 = 5,b0 = 5) and a PI+CI compensator (kp = 5.5,

τi = 0.8) with variable band resetting law (θ= 0.1). . . 179

5.7 Output signals, control signals, and reset ratio of the reset control system and its base PI from the Example 5.2 for a sequence of step setpoints. PI (dotted-blue), PI+CI with zero crossing resetting law (dashed-green) and PI+CI with variable band resetting law (solid-black). . . 181 5.8 Output signals, control signals and reset ratios of the reset control

system from the Example 5.2 with a 10% of uncertainty in the plant gain. k= 0.9 (dashed),k = 1 (solid), and k = 1.1 (dotted). . . 182

5.9 Output signals, control signals, and reset ratio of the reset control system and its base PI from the Example 5.2 for a step disturbance. PI (dotted-blue), PI+CI with zero crossing resetting law (dashed-green)

and PI+CI with variable band resetting law (solid-black). . . 184 5.10 PIbase stabilizing region in the (ˆkp,ˆki)-plane (shadow area), and values

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List of Figures ix

5.11 Output signals and control signals for several PI and PI+CI compensat-ors applied to an IPDT process with k = 0.2 and h= 7.4. . . 196

5.12 Output signals and control signals for several PI+CI compensators and its base compensators applied to an IPDT process with k = 0.2 and h= 7.4. . . 197

5.13 Performance indices for several PI and PI+CI compensators applied to an IPDT process (k = 0.2 and h = 7.4) with a 15% of uncertainty in

the gain and the time-delay. . . 199 5.14 Food industry pilot plant for pH control experiments. . . 203 5.15 Output signals (pH value), control signals (wpm), and reset ratio for

a PI and a PI+CI applied to the in-line pH process. PI (dotted) and PI+CI (solid). . . 205 5.16 Liquid level process. . . 206 5.17 Output signals (water level) and control signals (percentage of the pump)

for a PI and a PI+CI applied to the liquid level process with a sequence of step setpoints. PI (dotted) and PI+CI (solid). . . 209 5.18 Output signals (water level) and control signals (percentage of the pump)

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List of Tables

1.1 Publications in the research field of reset control. Journal (J), conference (C), thesis (T), and book (B). . . 15

2.1 MatricesA,Ad,C, andCh of a time-delay reset control system for input

and output time-delay. . . 48 2.2 PI+CI configurations for the Example 2.3. . . 61

3.1 Maximum allowable input/output time-delay,hM, for the Example 3.5

obtained by Corollaries 3.2, 3.3, and 3.4. . . 90 3.2 Maximum allowable input/output time-delay,hM, for the Example 3.6

obtained by Corollaries 3.2, 3.3, and 3.4. . . 91 3.3 Maximum input/output time-delay, for the Example 3.6 obtained by

Proposition 3.7. for several values of the parameterr. . . 107

3.4 Possible combinations in equation (3.178). . . 111 3.5 Possible combinations in equation (3.167). . . 113 3.6 Maximum allowable input/output time-delay for the time-delay reset

control system of the Example 3.10 obtained by Proposition 3.9. . . 113

5.1 Compensators setting and performance indices for the Example 5.2. . . 180 5.2 Compensators setting and performance indices for the Example 5.3. . . 183 5.3 Compensators setting and performance indices for an IPDT process

with k = 0.2 and h= 7.4. . . 195

5.4 Worst-case performance indices for several PI and PI+CI compensators applied to an IPDT process (k = 0.2 and h = 7.4) with a 15% of

uncertainty in the gain and the time-delay. . . 198 5.5 Tuning rules for first/second order processes and IPDT processes. . . . 201 5.6 Compensators parameters and performance indices for the in-line pH

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xii List of Tables

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Chapter 1 Introduction

In this chapter, we give an introduction to reset control systems. First, the reset control is motivated by the overcoming of the fundamental limitations of the linear control. Second, it is provided a survey of reset control raging from the beginning in 1958 to 2015. In addition, we perform an analysis of the bibliography in the field of reset control systems, providing a complete list of the published works, and interesting information such as the number of published works by year and the distribution by countries. Finally, we expose the motivations and the main objectives of the thesis.

1.1 Preliminaries

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2 Chapter 1. Introduction

Figure 1.1 Standard feedback control system formed by a plant P and a compensator C.

limitation on the feedback control.

A standard feedback control system is showed in Fig. 1.1 and consists of the feedback interconnection of a system P (normally called plant) and a compensator1 _C,

that implements the control strategy. The compensator senses the operation of the system by a measured signaly (probably affected by asensor noise n), compares it

against the desired behaviorr (reference signal), computes corrective actionsu (control

signal) based on a model of the system, and actuates the system to effect the desired change. Here, the system P is also influenced by an exogenous signal d (disturbance

signal). The control system can be simply modeled using Laplace transform, when both the plant and the compensator are linear and time-invariant (LTI) systems. The function L(s) =P(s)C(s) is the (open) loop transfer function, where P(s) and C(s)

are the transfer functions of the plant and the compensator respectively. Since the system is LTI, the relations between the inputs and the exogenous signals can be expressed in terms of transfer functions. The feedback system in Fig. 1.1 is completely characterized by four transfer functions, called the Gang of four ([9]):

• Complementary sensitivity function: T(s) = ₁₊L_L(s₍)_s₎.

• Load disturbance sensitivity function2: P(s)

1+L(s).

• Noise sensitivity function3: C(s)

1+L(s).

1_{The difference between compensator and controller is fuzzy. Both terms can be found in the}

literature without any distinction between them. However, some authors reserve the term compensator to those controllers that can be defined in the form of rational transfer functions. In this way, the term is linked to the classic control design of lead and lag phase compensation. In this thesis, both terms will be used interchangeably, although compensator will be used preferentially.

2_{The load disturbance sensitivity function is also called the input sensitivity function.}

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1.1 Preliminaries 3

• Sensitivity function: S(s) = ₁₊1_L₍_s₎.

The feedback control system can be asked to satisfy many performance specifica-tions, which clearly lead into a trade-off between the multiple performance objectives. This constitutes a classical and important issue in control theory, which is the topic of fundamental limitations in feedback systems. This line of work for linear systems traces back to the seminal work of H. W. Bode [51] in 1940.

For a large class of practical feedback control systems (with no open-loop zeros and poles in the right half plane, and enough roll-off), Bode showed that the logarithmic sensitivity integral is zero, that is

Z ∞

0 log

|S(jω)|dω = 0, (1.1)

which means that making the sensitivity magnitude small at some frequency range necessarily makes the sensitive magnitude larger at other frequency range. This prin-ciple is known as Bode’s integral formula, and establishes the bases of the fundamental limitations of the LTI systems (see, e.g., [248]). For instance, a consequence of the Bode’s integral formula is that if disturbance attenuation is improved in one frequency range, typically at low frequencies, then the appearance of disturbance amplification is unavoidable at higher frequencies. I. M. Horowitz ([148]) used Bode’s results in feedback control design, given also some preliminary results about the logarithmic sensitivity integral for open-loop unstable systems. Several other works investigated fundamental limitations in the presence of open-loop systems with poles and zeros in the right half plane, and also with time-delays ([98, 100, 101]). Some fundamental limitations are inherently linked to the plant ([93]). However, other limitations are consequence of using LTI control, and thus, in principle it might be possible that more complex control strategies, such as nonlinear control, overcome these limitations and outperform all the linear control design solutions. This is the main motivation of using non-LTI compensation.

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0 2 4 6 8 10 12 14 16 18

−2 −1 0 1 2

Time

Figure 1.2 Response of the Clegg integrator and the linear integrator to a sinusoidal input. Input (dotted), linear integrator (dashed), and Clegg integrator (solid).

counterpart, using a describing function analysis (see, e.g., [107]). The CI describing function is given by

CI(jω) = 1.62 jω e

−j38.1◦ _(1.2)

and has the distinctive characteristic of having only 38.1 degrees of phase lag compared to the 90 degrees of an integrator, which gives an extra phase lead of 51.9 degrees at every frequency. An, in addition, the magnitude is only increasing by a factor of 1.62, that is 4.19 dB. Although this analysis should be taken with some care, due to the fact that the describing function analysis can only be used in a qualitative way, this extra phase lead of 51.9 degrees at all frequencies supposes a breakdown of the stringent relationship (1.1), making possible to obtain values R∞

0 log|S(jω)|dω < 0.

For example, for a control system with plant P(s) = _s₊₁1 and a CI as compensator

C, by using (1.2) it is obtained that R∞

0 log|S(jω)|dω ≈ −0.85, thus CI overcomes

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1.2 Survey of Reset Control 5

Besides the CI, several other nonlinear compensators appeared in the following years, aim at overcoming fundamental limitations of LTI control; for instance the driven limiter [11], the split-path nonlinear (SPAN) filter [97], and the nonlinear integrator with no phase-shift [165]. It turns out that these nonlinear compensators, including CI, were proposed in a rather heuristic way, waiting for rigorous control theory to come. It was not until impulsive and hybrid systems theory became mature that these control design techniques started to be seen as a serious alternative to more traditional LTI compensation.

1.2 Survey of Reset Control

Reset control began undoubtedly with the appearance of the Clegg integrator in the work of Clegg [79]. However, it was not until the works of Horowitz and coworkers that reset control was quantitatively incorporated into control design. First, Krishnan and Horowitz [168] proposed a complete and quantitative control design procedure based on the Clegg integrator. They provided the feedback control system with a reset structure consisting of a Clegg integrator in parallel with a linear integrator. The favorable properties of the describing function of the Clegg integrator permit a faster reduction of the loop magnitude, and thus, the sensor noise amplification4. Subsequently, Horowitz

and Rosenbaum [151] extended the analysis and design procedure to a first order reset element (FORE). The key point in the design was a two step procedure: first, a linear compensator is designed with the aim of meeting all the design specification except for the overshoot, then the pole of the FORE is selected to meet the overshoot specification.

In spite of the fact that the design procedure in the works of Horowitz did not have recourse to the describing function or any other approximation analysis, their research was restricted to the CI and FORE elements, and no general theory was developed. Most likely for this reason, research on reset control systems was not further developed during two decades, until impulsive and hybrid dynamical systems theory became a mainstream research line. The conclusion of the seminal work [151] ends up with the sentence:

4_{The sensor noise amplification is one of the fundamental limitation of the feedback. The sensor}

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“This problem will be solved more satisfactorily when stability criteria are developed for feedback loops containing non-linear elements such a FORE”.

In 1995, J. E. Bobrow, F. Jabbari and K. Thai proposed in [50] the use of a resetting action to suppress mechanical vibrations of flexible structures. The work does not seems to be influenced by the previous works due to the lack of references to them. In addition, it can be considered as the first experimental application of reset control. Two years later, two new works on reset control appeared, [160] and [58], and they supposed the new dawn of reset control research. The work of H. Hu, Y. Zheng, Y. Chait and C. V. Hollot [160] is the first of a large number of journals, conference proceedings, and several PhD thesis developed by the authors and coworkers. The development of a theoretical framework for general reset control (disseminated through their works, see for instance [47]) can be considered as one of their main contribution on the field of reset control. They also produced a series of papers exploring stability analysis of reset control systems [45, 48, 74, 75], performance analysis [72, 73, 144], experimental application of reset control [306], and demonstration of reset control overcoming fundamental limitations of linear feedback [46]. Their intensive research culminated in one of the most important stability results on reset control systems, the so-called Hβ-condition[47]. Despite these works consider a more general class of reset

control systems with higher order compensators, they keep intact the original ideas of resetting the compensator states to zero when its input is zero. This characteristic is the hallmark that has made the reset control different from other related approaches, such as impulsive/hybrid control systems.

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framework for the design of energy-dissipating hybrid controllers ([138]).

Although there were different research lines regarding reset control during the 1990s and mid 2000s, in general, they all adopted the mathematical formulation of impulsive differential equations, mainly developed during the decade of the 1980s in Russia (see, e.g., [12, 172], and the references therein). However, L. Zaccarian, D. Nešić and A. R. Teel exposed in [288] that the zero crossing formulation for reset control systems developed by Y. Chait, C. V. Hollot, and coworkers is prone to problems. They asserted for instance that:

• It does not represent the behavior of the circuit implementation of the Clegg integrator ([79]) for all the initial conditions.

• It does not provide robust results, for example robust stability, mainly since the reset condition is defined by a hyperplane, and thus, it is possible to generate arbitrarily small noise that will prevent the state from resetting.

• The trajectory of the system can flow overall the state-space except in a set of zero measure (a hyperplane). Consequently, the system is required to satisfy suitable conditions on the whole state-space for establishing useful stability results (see [47]).

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been developed in the field of reset control systems based on this new hybrid model approach: stability analysis in [143, 209, 211, 236, 237, 289, 291, 299], performance analysis in [1, 2, 190, 276, 290], saturated reset control systems in [189, 262], experi-mental applications in [222, 223], reset passivation in [96], etc.

In mid 2000s, another line of research, essentially practical, emerged in the field of reset control (see, for example, [108, 109, 130, 174, 177, 303, 304]). This line is mainly characterized by the following two features:

• The reset actions occur at some fixed instants that are previously design, for instance the reset instants can be periodic.

• The states are not simply reset to zero. Then the problem is how to design the sequence of reset actions, which is normally design to minimize some cost function.

This approach has been successfully applied to the control of hard disk drive (HDD) servo systems in [130, 174, 177, 181] and to the control of a piezoelectric (PZT) micro-actuator positioning state in [304].

Alongside the above approaches, the initial representation of reset control system as impulsive dynamical systems is further developed since 2006, mainly by the works of A. Baños, A. Barreiro, and coworkers. By the first time, reset control systems were applied to systems with time-delays; since past experience shows that one of key instruments in reset compensation is the extra phase lead obtained with respect to its base5 _{LTI compensator, application to systems with time-delays seems to be natural}

field since, in principle, phase lead can be used to alleviate the phase lag due to the time-delay in a feedback loop. Several research lines related with reset control has been developed, not only related to time-delay systems (most of the work is summarized in the recent monograph [20]):

• A dependent stability analysis is accomplished in [37], providing delay-dependent conditions in both the time domain and frequency domain; in addition, delay-independent stability analysis and a generalizedHβ-condition for time-delay

reset systems, based on a generalized Kalman-Yakubovich-Popov (KYP) lemma, is developed in [19]. This research line has been subsequently followed in several works ([82, 133, 196, 197, 236]), being one the main roots of this thesis work.

5_{In absence of reset actions, the resulting linear system is called the base linear system, or in}

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• Formalization of the impulsive dynamical systems framework, avoiding the use of time regularization, basically searching for well-posedness and stability criteria that gives a solid foundation to reset control systems. It has been shown that for exogenous signals with some smoothness properties (Bohl functions), reset control systems are well-posed (existence and uniqueness of solutions and continuous dependence on initial conditions) for most of the reset compensators that has been used in practice ([28, 29]). Stability results for stable and unstable base systems by time-dependent conditions on the reset instants are given [22, 23], improving Hβ-condition (only usable for some stable base systems); in addition,

input-output stability has been also approached in [66], based on application of passivity results; successfully applied to passive teleoperation in [91, 92].

• Development of PI-based reset compensators, with the goal of obtaining a simple and easy-to-tune reset compensator, with special application to process control. The result has been the PI+CI compensator, with different resetting laws: zero crossing, fixed reset band, variable reset band, anticipative reset ([34, 35, 270, 271]), that has been successfully applied in a number of practical

cases ([25, 83, 271, 272]).

Other research lines in the field of reset control systems, directly or indirectly related with the above impulsive dynamical systems approach, are:

• Reset observers: This line of research began in 2010 with the work [219],

motivated by the advantageous properties that reset elements have shown in control application. In that work, the authors proposed a new adaptative observer called reset adaptative observer, where a linear integrator is substituted for a reset element. This strategy supposed a breakdown of the limitation of linear adaptative observer to reduce simultaneously the overshoot and settling time of the estimation process. Further developments in this line can be found in: optimal observer design [216], MIMO systems [220], nonlinear systems [119, 221] and time-delay systems [217, 294]. It is interesting to mention that reset adaptative observer has been successfully used to control the cooking pot temperature in induction hobs (see [215]).

• Discrete-time reset control systems: Nowadays, analog controllers have

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that are adequate to computer-based implementation. Discrete time-models of reset control systems were used in [174, 176, 177] to applied reset compensation to control servomechanisms in hard disk drives. However, it is not until the work [31] that a formal representation of time-discrete reset control systems, coherent with the existing continuous-time results, is developed. Discrete-time reset control under this formulation has been applied to networked control systems (NCS) in [30, 32, 228, 229]. The works [123, 124, 284] are concerned with the stability analysis and design of reset control systems when the reset condition is replaced by a discrete-time counterpart. Other works in the line of discrete-time reset control systems are [67, 134].

• Fractional-order reset control systems: In this line, fractional dynamics of

hybrid systems is studied [156]. In particular, properties of reset control systems with fractional-order are analyzed as a subclass of the fractional hybrid systems [153, 154]. The applicability of fractional-order reset control is illustrated in [155] with an example of speed control of a servomotor. Most of the results in this line can be found in [152, 157, 158].

Throughout reset control ages, a multitude of experimental application have shown the potentials benefits of reset compensation, starting from the control of mechanical elements until the process control application in the last years. A sample of the many applications is:

• Vibration suppression for flexible structures: [50].

• Tape-speed control system: [306].

• Speed control of rotational flexible mechanical system: [144].

• Hard disk drive: [125, 126, 130, 131, 146, 174, 176, 177, 181, 279].

• Marine thruster control: [15].

• Piezoelectric actuator positioning: [303, 304].

• Teleoperation: [89, 91, 92].

• Solar collector fields: [272].

• In-line pH process: [25, 62, 230].

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• Liquid level control: [83].

• Exhaust gas recirculation valve: [222, 223].

• Induction hobs: [215].

• Gantry crane: [240].

• Servomotor-speed control: [155].

• Synchronous motor with permanent magnet excitation: [86] (simulation).

1.2.1 Analysis of the Bibliography

In spite of the sound previous works and the increasing interest of the control community in the last years, reset control system is still in infancy. If a bibliographic search is performed, after some treatment it can be observed that there are no much more than a hundred published works (including all the conferences and journal papers that usually have some similar contents), which is a clear indicator of the initial development of the subject. It is interesting to see some figures about the state of the art in reset control systems6_{. In Table 1.1, it is shown authors, date an type of publication; Fig. 1.3 shows}

a historical view of the number of published works by year, and Fig. 1.4 a distribution by countries (thesis and books have been excluded). The H-index of the topic “reset control” is 20 (for the provided list of works).

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1958 1974

1975 1977

1995 1997

1999 2000

2001 2002

2004 2005

2006 2007

2008 2009

2010 2011

2012 2013

2014 2015 0

5 10 15 20 25 30

Figure 1.3 Historical view of the number of published works by year.

Spain United States Singapore China Australia Italy France United Kingdom Netherlands Portugal Austria Brazil Iran Japan Mexico Norway South Africa Taiwan

0 10 20 30 40 50 60 70 80

[image:40.595.121.478.419.678.2]

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Authors Date Ref. Clegg 1958 J [79] Khishnan, Horowitz 1974 J [168] Horowitz, et al. 1975 J [151] Karybakas 1977 J [165] Bobrow, Jabbari, et al. 1995 J [50] Hu, Zheng, et al. 1997 C [160] Hollot, Zheng, et al. 1997 C [145] Bupp, Bernstein, et al. 1997 C [58] Zheng 1998 T [305] Beker, Hollot, et al. 1999 C [48] Beker, Hollot, et al. 1999 C [43] Bupp, Bernstein, et al. 2000 J [59] Zheng, Chait, et al. 2000 J [306] Chen, Hollot, et al. 2000 C [74] Haddad, et al. 2000 C [137] Chen, Hollot, et al. 2000 C [75] Chen, Chait, et al. 2000 C [72] Beker, Hollot, et al. 2000 C [44] Chen 2000 T [71] Chen, Chait, et al. 2001 J [73] Beker, Hollot, et al. 2001 J [46] Beker, Hollot, et al. 2001 C [45] Hollot, Beker, et al. 2001 B [144] Beker 2001 T [42] Chait, Hollot 2002 J [68] Beker, Hollot, et al. 2004 J [47] Zaccarian, Nešić, et al. 2005 C [288] Nešić, Zaccarian, et al. 2005 C [210] Li, G. Guo, Wang 2005 C [180] G. Guo, Yu, Ma 2006 J [121] Hong, Wong 2006 C [146] continue in the next column

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Authors Date Ref. Li, Du, Wang 2009 C [174] Li, Du, et al. 2009 C [177] Vidal, Baños 2009 C [270] Baños, Dormido, et al. 2009 C [26] Baños, Perez, et al. 2009 C [31] Li, Du, et al. 2009 C [178] K. El Rifai, O. El Rifai 2009 C [242] Raimundez, et al. 2009 C [239] Carrasco, Baños, et al. 2009 C [63] Y. Guo, Wang, et al. 2009 C [128] Carrasco 2009 T [61] Vidal 2009 T [268] Aangenent, et al. 2010 J [1] Zheng, Fu 2010 J [302] Carrasco, Baños, et al. 2010 J [66] Barreiro, Baños 2010 J [37] Vidal, Baños 2010 J [271] G. Guo 2010 J [120] Paesa, Franco, et al. 2010 C [219] Polenkova, et al. 2010 C [231] Loquen, Nešić, et al. 2010 C [188] Baños, Perez, et al. 2010 C [32] Li, Wang 2010 C [179] Loquen 2010 T [187] Zaccarian, Nešić, Teel 2011 J [291] Baños, Carrasco, et al. 2011 J [23] Fernández, Blas, et al. 2011 J [92] Paesa, Franco, et al. 2011 J [220] Baños, Dormido, et al. 2011 J [27] Tarbouriech, et al. 2011 J [262] Nešić, Teel, et al. 2011 J [209] continue in the next column

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Authors Date Ref. Panni, Alberger, et al. 2012 C [222] Y. Guo, Xie 2012 C [133] Davó, Baños 2012 C [81] Polenkova, et al. 2012 C [232] Baños, Barreiro 2012 B [20] Moreno, et al. 2013 J [202] Guillen-Flores, et al. 2013 J [119] Nešić, Teel, et al. 2013 J [207] HosseinNia, et al. 2013 J [155] Prieto, Barreiro, et al. 2013 J [235] Bras, Carapito, et al. 2013 J [55] Hetel, Daafouz, et al. 2013 J [143] Carrasco, et al. 2013 J [67] Liu, Xum Wang 2013 J [185] Lu, Lee 2013 J [191] Li, Fen, Xinmin 2013 C [173] HosseinNia, Tejado 2013 C [156] Falcon, Barreiro, et al. 2013 C [89] HosseinNia, et al. 2013 C [154] Wu, Guo, Gui, Jiang 2013 C [280] Davó, Baños 2013 C [83] Mercader, Davó, et al. 2013 C [197] Suyama, Kosugi 2013 C [260] Davó, Baños 2013 C [82] Mercader, et al. 2013 C [196] Zhao, Nešić, et al. 2013 C [292] Perez 2013 T [227] HosseinNia 2013 T [152] Yuan, Wu 2014 J [285] Ghaffari, et al. 2014 J [108] Yuan, Wu 2014 J [286] continue in the next column

HosseinNia, et al. 2014 J [157] Shakibjoo, Vasegh 2014 J [253] Panni, Waschl, et al. 2014 J [223] Ghaffari, et al. 2014 J [109] Zhao, Wang 2014 J [294] Ogura, Martin 2014 J [214] Baños, Davó 2014 J [25] Barreiro, Baños, et al. 2014 J [40] Baños, Perez, et al. 2014 J [30] HosseinNia, et al. 2014 C [158] Delgado, Cacho, et al. 2014 C [86] Y. Guo, Zhu 2014 C [134] Bragagnolo, et al. 2014 C [54] Zhao, Wang, Li 2014 C [299] HosseinNia, et al. 2014 C [153] Zhao, Nešić, et al. 2014 C [293] Davó, Baños, Moreno 2014 C [84] Acho 2014 C [3] Zhao, Wang, Li 2014 C [298] Vettori, et al. 2014 C [267] Delgado, et al. 2014 C [85] Prieto 2014 T [234] Baños, Perez, et al. 2014 B [33] Zhao, Wang 2015 J [297] Zhao, Wang 2015 J [295] Zhao, Wang 2015 J [296] Zhao, Wang, et al. 2015 J [300] Zhao, Yin, Shen 2015 J [301] Yu, Y. Guo, Zhao 2015 J [284] Heertjes, et al. 2015 C [141] Mercader, Davó, et al. 2015 C [198]

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1.3 Motivation and Objectives

1.3.1 Motivation

Since the work of Bode in 1940 [51], a multitude of researchers have brought to light the factors that fundamentally limits the achievable performance of linear control systems (see, e.g., [99, 248]). For instance, Bode’s gain-phase relationship assumes stable, minimum phase loop transfer function and links loop gain and phase. This relationship implies that if the loop phase is increased to provide large phase margin, and thus, good tolerance to plant uncertainty near the crossover frequency, then it will result in a slow gain attenuation. As a consequence, more measurement noise is injected into the system, producing a deterioration of the performance. Fundamental limitations have been also studied for non-minimum phase systems, poles and zeros in the right half plane and time-delay ([6, 100]), showing that there are more restrictive limitations in those cases. Unfortunately, time-delay is present in most of the industrial processes, so that the limitation of the LTI compensation is important beyond the theoretical realm. This motivates the challenging task of finding nonlinear compensator to be able to overcome the fundamental limitations of the linear control. Reset control systems appeared in 1958 with the invention of the Clegg integrator. Its advantageous properties suggested by the describing function made the CI a potential element to overcome the linear compensation. During the last 20 years a multitude of works have shown real benefits of the reset compensation over linear feedback control schemes, overcoming in some cases the fundamental limitations of linear control without resorting to approximations [46]; from theoretical results dealing with stability [23], asymptotic tracking [1], disturbance rejection [96, 291, 293], etc, to experimental applications [15, 202, 223, 272, 306].

It should be emphasized that many nonlinear compensators have been suggested in the literature, which exhibit similar advantages as reset compensation. However, as mentioned in [150] by the pioneer in the field of reset control Prof. Horowitz:

“The difficult challenge is how to integrate such nonlinear elements into a systematic quantitative design technique”.

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1.3 Motivation and Objectives 17

base linear system, and thus, easier to understand. On the other hand, conditional expressions and reset action of the linear integrator are, in general, needed for the implementation of reset compensators, and those features are common in most of the simulation softwares and industrial programable logic controllers (PLC). In this way, reset compensation could be seen as another tool available in the toolbox of the control engineers, such as two-degree of freedom scheme and anti-windup strategies, whose appropriate application on a linear feedback control scheme can lead into substantial benefits (the PI+CI compensator and the Smith predictor plus Clegg integrator com-pensator in [202] are a clear illustration of this idea).

Finally, it should be emphasized, the importance of the theoretical insight into the reset control systems to the more general field of impulsive and hybrid systems. Since the reset control systems are a subclass of the impulsive/hybrid systems, any particular result for reset control systems has the potential to be generalized and applied to those more general systems.

1.3.2 Objetives

Our target in this thesis is that providing advance in the field of reset control systems; both from the theoretical and engineering applications point of views. The general objectives of the thesis are exposed in the following subsections.

Stability Analysis of Time-delay Reset Control Systems

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main reason why the stability analysis of time-delay reset control systems is more challenging. Only few works7 have dealt with the stability analysis of time-delay reset

control systems ([19, 37, 133, 196, 236]), being the results conservative. The main limitation of these results is the need of a stable base system. The stabilization of unstable systems by impulsive actions has been addressed in several works in the field of impulsive systems (see, e.g., [256] and the references therein). However, most of the results are not focused on control problem, and they do not regard with the limitations imposed by the fixed structure of a control closed-loop system. Consequently, to the best of the author knowledge, they are not applicable, at least directly, to time-delay reset control systems.

In this context, one of the main objective of this thesis is to developed new stability criteria that: in first place, can guarantee the stability of the time-delay reset control system for time-delays greater than those obtained by the results in the literature, and second, completely remove the restrictive condition of a stable base system.

Stability Analysis of Reset Control Systems with Saturation

Saturation is one of the most common nonlinearities encountered in control practice, since many physical systems are subject to magnitude-limitation in its input. This may cause performance deterioration, limit cycles, and in some cases unstable behavior. Investigation of how saturation may impact control systems stability and performance has attracted research efforts from several decades (see, e.g., [163, 261], and the references therein). A key topic, that has received considerable attention, is the estimation of the region of attraction. The analysis of reset control systems in presence of saturation has been tackled for the first time in [189], considering the FORE compensator, and using Lyapunov-like approaches and a hybrid systems framework. Some extensions, in the same context, have been proposed in [94, 262]. Another goal of this thesis is the analysis of how reset compensation can alleviate the effect of saturation on the closed-loop stability, focusing on the development of a procedure to estimate the region of attraction.

New Tuning Methods for the PI+CI Compensator

Proportional integral (PI) and proportional integral derivative (PID) compensators are by far the most adopted control strategies in industry. It is mainly owing to is

7_{The analysis of discrete-time time-delay reset control systems has been accomplished in [30].}

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1.4 List of Publications 19

simplicity and capability of solving a wide range of control problems. In addition, a great effort has been made to fulfill the need of methods for finding good values of the compensator for different processes, making the PID compensator accessible to control engineering practitioners. The PI+CI compensator is devised in [35] as a reset counterpart of the PI, showing its advantageous over the PI compensator in a multitude of works. In the attempt to provide a higher improvement of the reset compensation, several modifications of the PI+CI has been proposed in the literature. However, the lack of systematic procedure for the tuning of the PI+CI compensator complicates its applicability. In this context, another objective of the thesis is to develop a systematic method to successfully design a PI+CI compensator, resulting whenever it is possible, into simple tuning rules for its design parameters.

Experimental Application of the Proposed Tuning Rules

As a final objective, it is considered the application of the PI+CI with the developed tuning rules to real processes, in order to illustrate the industrial applicability of the results. Two industrial control processes are considered as a target: in-line control pH processes and liquid level control processes. In the first case, the experiments will be carried out in an industrial pilot plant, design for the experimentation of thermal treatments common in the tinned food industry. In addition, the pilot plant will be provided with a water tank in order to perform the liquid level experiments.

1.4 List of Publications

Most of the results exposed in this thesis have been published (or submitted) to several international conferences and journals. In this section we provide the list of publications:

• Tuning rules for a reset PI compensator with variable reset. A. Baños and M. A. Davó. Proceedings of the IFAC Conference on Advances in PID Control (2012), [24].

• PI+CI tuning for integrating plus deadtime systems. M. A. Davó and A. Baños. Proceedings of the 17th _{Conference on Emerging Technologies & Factory}

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• Reset control of a liquid level process. M. A. Davó and A. Baños. Proceedings of the 18th Conference on Emerging Technologies & Factory Automation (2013),

[83].

• H∞/H2 analysis for time-delay reset control systems. P. Mercader, M. A. Davó

and A. Baños. Proceedings of the 3rd _{International Conference on Systems and}

Control (2013), [197].

• Delay-dependent stability of reset control systems with input/output delays. M. A. Davó and A. Baños. Proceedings of the 52nd _{Conference on Decision and Control}

(2013), [82].

• Tuning of reset proportional integral compensation with a variable reset ratio and reset band. A. Baños and M. A. Davó. IET Control Theory and Applications

(2014), [25].

• Region of attraction estimation for saturated reset control systems . M. A. Davó, A. Baños and J. C. Moreno. Proceedings of the 14th _{International Conference on}

Systems, Control, and Automation (2014), [84].

• An impulsive dynamical systems framework for reset control systems. A. Baños, J. I. Mulero, A. Barreiro and M. A. Davó. ePrint arXiv: 1505.07673 (2015), [29] (submitted to International Journal of Control).

• Performance analysis of PI and PI+CI compensation for an IPDT process. P. Mercader, M. A. Davó and A. Baños. Proceedings of the 23rd _{Mediterranean}

Conference on Control and Automation (2015), [198].

• Stability of time-delay reset systems with a nonlinear and time-varying. A. Baños and M. A. Davó. International Journal of Robust and Nonlinear Control

(submitted).

• Reset control of integrating plus dead time processes. M. A. Davó and A. Baños.

Journal of Process Control(second revision).

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1.5 Overview of Contents 21

1.5 Overview of Contents

This thesis is organized as follows:

• In Chapter 2, the description of the framework of impulsive dynamical systems (IDS) and its extension to impulsive delay dynamical systems (IDDS) is given. It is also provided a definition of reset control systems and time-delay reset control systems under the IDS (IDDS) framework. Several results on the well-posedness of the reset control systems are also given. The PI+CI compensator is introduced as a particular reset compensators, and a multitude of design improvements are analyzed. A framework for hybrid systems and its extension for hybrid systems with time-delay are also presented.

• In Chapter 3, we deal with the stability analysis of impulsive delay dynamical systems. Firstly, a general Lyapunov-Krasovskii-based criterion is devised for the stability of state-dependent IDDSs with nonlinear and time-varying base systems. The general result is applied to several LK functionals to developed stability criteria in the form of LMIs for checking the stability of an IDDS with nonlinear and time-varying base system and a particular structure, and an IDDS with LTI base system. In the second part of the chapter, two general Lyapunov-Krasovskii-based criteria are obtained based on time-dependent conditions. The first result provides less conservative conditions for IDDSs with stable base LTI systems and the second one allows the analysis of IDDSs with unstable LTI base systems. Finally, it is obtained LMI conditions for the stability of a time-delay reset control system comprises a LTI plant and a PI+CI compensator.

• In Chapter 4, we focus on the influence of the saturation on the reset control system. In particular how saturation may impact the region of attraction of the control system. The main idea is to represent the set of trajectories of the closed-loop reset control system as a directed graph of polytopes and arcs between them. Several conditions of positive invariance and reachability of the origin for the polytopes are used to analyze the graph and compute an estimate of the region of attraction. The proposed technique is compared with the method proposed in the literature.

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rules are provided for first/second order plants and integrating plus dead time plants. Finally, it is presented the experimental results obtained when applying the PI+CI with the proposed tuning rules to an in-line pH control process and a liquid level control process.

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Chapter 2 Reset Control Systems

In this chapter, we give basics of reset control systems. The framework of impulsive dynamical systems and its extension of impulsive delay dynamical systems are used for the description of the (time-delay) reset control systems. Fundamental results regarding the well-posedness (existence, uniqueness and continuous dependence on the initial condition) of the reset control systems are provided. In addition, other frameworks for the modeling of reset control systems are presented. Finally, a proportional-integral plus Clegg integrator (PI+CI) compensator is introduced, considering several design improvements proposed in the literature. In addition, new modifications with potential benefits in practical applications are proposed.

2.1 Preliminaries

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24 Chapter 2. Reset Control Systems

piecewise affine models, hybrid Petri nets, hybrid inclusions, etc. Models for continuous systems and for discrete systems have been originally studied by different disciplines, such as control theory (continuous dynamic) and computer science (discrete dynamic). Consequently, the different frameworks for hybrid systems analysis and modeling place greater emphasis on the continuous or discrete dynamic depending on their discipline origin. An important class of hybrid systems are the impulsive systems, which are characterized by a really simple discrete part and a complex continuous dynamics. The discrete dynamic of the impulsive systems arises from instantaneous changes of the state variables that occur when a simple condition is satisfied, for example some prefixed instants are reached, or the trajectory hits a predefined hypersurface. It is clear that reset control systems lay into the class of impulsive systems, and in general hybrid systems. The rich continuous behavior of the impulsive systems, and in particular reset control systems, makes some frameworks more adequate to model and analyze this kind of systems. Therefore, in spite of the wide variety of frameworks for hybrid systems, in this chapter, it will be only introduced those frameworks1 _{that have been}

successfully used for modeling reset control systems.

An important issue in the study of impulsive/hybrid systems, directly related with their solution concept, is well-posedness. Historically, the term well-posedness dates back to 1902 when Jacques Hadamard ([136]) formulated three conditions that mathematical models of physical phenomena should satisfy, namely that:

1. A solution exists.

2. The solution is unique.

3. The solution depends continuously on the problem data.

An impulsive/hybrid system with the above properties will be denoted as a well-posed

system. The posible refinements of the notion of well-posedness to be made throughout the document will be explicitly mentioned.

Impulsive/hybrid systems involve an interacting mixture of continuous and discrete dynamics, which provides the systems with a very rich dynamical behavior. In particular, the trajectories of impulsive/hybrid systems can exhibit multiple complex phenomena such as:

1_{Reset control systems have been also modeled as a hybrid automata (see, e.g., [194] for details on}

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2.2 Impulsive Dynamical Systems 25

• Deadlock: This situation corresponds to a dynamical system state from which no continuation, continuous or discrete, is possible.

• Beating2_. _{An impulsive/hybrid system experiences beating when the system}

trajectory encounters the same resetting surface a finite or infinite number of times in zero time.

• Zeno3 _{solutions: An impulsive/hybrid system possesses a} _{Zeno solution} _{if the}

system experiences an infinite number of discrete transitions in a finite amount of time.

• Confluence: Finally, this situation involves system solutions that coincide after some time and behave as a single solution thereafter.

These phenomena, along with other difficulties such as discontinuous solutions, make the analysis of the well-posedness of impulsive/hybrid systems extremely challenging.

2.2 Impulsive Dynamical Systems

One of the main frameworks for reset control systems modeling is the framework of

Impulsive Dynamical Systems4 _{(IDS) [138], which is mainly based on the theory of}

impulsive differential equations ([12, 172]). In this way, the description of the impulsive system is accomplished by an state-space representation, which is mainly composed by the following three elements:

• Differential equation. The continuous-time dynamic of the system between reset events is governed by a differential equation, as follows:

˙

x(t) =f(t,x(t)). (2.1)

In general, the function f can be both nonlinear and time-varying.

2_{Beating is also known as}_{Pulse Phenomenon}_{(especially in impulsive systems) and also as}_Livelock_,

mainly in the field of computation science.

3_{The name Zeno refers to the Greek philosopher Zeno of Elea (ca. 490-430 BC), who is best known}

for his paradoxes, designed to support that the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. An example is Zeno’s Second Paradox of Motion, in which Achilles is racing against a tortoise.

4_{The notation, definitions, and in general the results in [138] are adapted for a better correspondence}